Not sure if this question is still live--anyway, i just saw it so i'll assume it is.
I think your discomfort is well-grounded. A minute or two of theory might help before discussing what to do next. Based on the provisions you've read to me, the draft Agreement co-mingles two distinct (but related) concepts of intellectual property law: (i) ownership (of the intellectual property; and (ii) access/control of the physical article that embodies it.
An example: you go to Barnes & Noble and buy the novel "Snow Crash" by Neil Stephenson. Do you own that book? Sure. Does the purchase grant you unrestricted access (control) to that book? Of course. Can you move to Hollywood, write a screenplay based on the book, and sell it to a studio for millions? No.
The answers are different because different legal rubrics are involved. My point here is that ownership (of the intellectual property) and access (to the physical article, or the "implementation" of the IP, as it were) are not the same, and having a right to one doesn't (necessarily) confer a right to the other.
How's this relevant to your question? Well, the company that created the Spec is obviously worried about a single specific scenario: you receive their Spec, read it carefully, tell them "no thanks" then send the spec to your in-house developers or a cheaper outside vendor, who then develop the product described in the Spec. (I am not even remotely implying you would do that--just saying that's what they fear.)
Now, the key point is that this quite justifiable fear can be completely ameliorated by an Agreement that relates solely to access/control--it (usually) need not bother with (IP) ownership. (This is obvious the more you think about it--the purpose of the Agreement is to restrict what you can do with the Spec; it is not an IP license.) For example, one provision might forbid you from disclosing any portion of that Spec to anyone outside your own company for any reason (subject to the usual boilerplate, e.g., unless they breach the contract and you have to find another outfit to complete the work, etc.) A second provision might prevent you from disclosing the Spec to anyone within your company, prior to execution of a production contract, unless that person is necessary for evaluation of the Spec. (You mentioned your firm is small, so this moght not work, but you get the idea.) Finally, a reasonable and fair NDA might have a provision which forbids you from making any copies, in whole or in part, whether digital or otherwise, etc., etc. and upon evaluation of the Spec, you must return the all (numbered) copies to the company that created it. (Obviously all of these can be circumvented, but they work together, plus, developers are perhaps the most ethical group of people i know--if they see a label on a document that says "DO NOT COPY"--i don't think they will.)
So where does this leave us? Well, we haven't discussed "ownership" (of the IP). The reason we haven't discussed it is because it isn't necessary to provide the protection the other Company actually seeks. So, i don't think that provisions like "if [we] can establish by reasonable proof...." belong in a document of this sort--granted, they are often (usually) there, but they shouldn't be. Another reason IP ownership provisions should not be in the Agreement is that (and this is the source of your concern i suspect) the IP embodied in the Spec does not belong exclusively and completely to the other company, and clearly they know that. What belonged to you before you ever saw that Spec should not suddenly become theirs unless you can supply reasonable proof that you own it--that's absurd.
In sum, you might consider giving them protection against the risk they fear--but that doesn't mean you need to sign that document they've given you. My point is that an Agreement with provisions relating solely to access and control of the physical article (as in the preceding paragraph) is often sufficient to provide that protection.